Characterization of diffusing sub-10 nm nano-objects using single anti-resonant element optical fibers

Accurate characterization of diffusing nanoscale species is increasingly important for revealing processes at the nanoscale, with fiber-assisted nanoparticle-tracking-analysis representing a new and promising approach in this field. In this work, we uncover the potential of this approach for the characterization of very small nanoparticles (<20 nm) through experimental studies, statistical analysis and the employment of a sophisticated fiber and chip design. The central results is the characterization of diffusing nanoparticles as small as 9 nm with record-high precision, corresponding to the smallest diameter yet determined for an individual nanoparticle with nanoparticle-tracking-analysis using elastic light scattering alone. Here, the detectable scattering cross-section is limited only by the background scattering of the ultrapure water, thus reaching the fundamental limit of Nanoparticle-Tracking-Analysis in general. The obtained results outperform other realizations and allow access to previously difficult to address application fields such as understanding nanoparticle growth or control of pharmaceuticals.


SI 1: Design of the chip
The key for measuring small particles is to reduce the background signal as much as possible. It is important to note, that even the slightest reflection of the illumination into the objective is very intense compared to the elastically scattered light of nanoparticles. Even rays that have been reflected several times must not enter the lens or the field of view. This is done by avoiding background in the first place and directing unavoidable background away from the field of view. In addition, everything should have the same refractive index of fused silica, to prevent reflections in unwanted directions. To fulfill these requirements, we have developed our own chip as a specimen holder (see Fig. S-1).
The first step is to minimize the amount of light which is not in the measurement channel. We optimized the efficiency of the butt coupling, by using a delivery fiber with an output mode as close to the fundamental mode of the SEF as possible, which has a large diameter (dc = 17 µm) and very low NA (0.024). This is done by a single mode fiber (S405-XP, Thorlabs) with a customized fiber (single mode; NA = 0.05; core = 7µm) spliced to its end. Nevertheless, some light will be outside the ARE inside the glass. Thus, we apply a drop of index matching gel (G608N3, Thorlabs) on the SEF after the butt coupling. We added some soot to the gel to absorb the outcoupled light. This step is required for gold particles of 20 nm diameter and below.
The SEF is covered by a coverslip made of fused silica (d = 400 µm). The coverslip is horizontally aligned by two empty SEF on each side of the filled SEF. A glycerin-watermixture ( ( = 532 , 20°) = 1.46) is used as an index matching gel under the coverslip. The gel has a high viscosity and a convenient contact angle, so that it flows quickly under the coverslip without draining off.
Thus, the scattered light from the nanoparticles leaves the fiber without significant aberrations. In addition, the index matching gel increases the losses of the higher modes so that this light leaves the fiber.
To get this light away from the field of view, the SEF is placed on a customized fused silica block, which the outcoupled light can enter without any reflection or refraction and is visible as a cone around the SEF. Although most of the light will leave the glass block, some will be reflected by the glass-air-surface. To avoid stray light into the fieldof-view, the glass block is polished on every side, so that the reflected light beam goes further down the block. But it cannot reach the field-of-view, because a slit is sawed into the block and filled by cardboard.
The objective is placed around 4.5 mm behind the edge of the coverslip. This is the position with the lowest background.
The output of the SEF is sealed by the self-made index matching gel. The fiber end needs to be closed so that no flow occurs inside the channel caused by evaporation of the water. Using the gel is convenient, because it allows to image the fiber mode by attaching a coverslip to it.

SI 2: Diameter analysis
The diameter determination procedure employs mean squared displacement (MSD) analysis for each particle individually and is defined by: Theory: MSD( ): =< ( ( ) − ( + )) 2 > = 2 ⋅ ⋅ (1) Experiment: with lag-time , lag-frame i, trajectory length f , diffusion coefficient D, drift velocity v and localization accuracy 2 (< > is the expectation value operator). The expectation value is approximated by averaging over all measured displacements given by the measured trajectory. The drift velocity v should be 0 at any time in the experiment.
Then, the diffusion coefficient can be retrieved as the slope of the MSD curve. The ideal number of considered lag-frames i is set by the experimental parameters 1 .
The NP diameter is retrieved by the Stokes-Einstein equation considering the hindrance factor, which is yet close to 1 2 . Note that the more frames f a continuous trajectory consist of, i.e., a NP is tracked, the more accurate is the approximation in Eq (2), thus leading to a lower statistical error D . Furthermore, the error is influenced by the localization accuracy and the particles motion blur, leading to a total minimum relative error of 1,3 : Eq (2) and (3) represent the key advantages of NTA: If a particle can be tracked, its individual diameter can be retrieved and the precision can be tuned by the number of frames f the movement is captured. The CV is a standardized measure to quantify the precision of a statistically limited measurement process such as NTA.
The rel. error of induced by the drift can be expressed by the drift induced displacement drift = ⋅ and the expected brownian motion < brown >= √2 ⋅ ⋅ between two frames: The rel. error is a quadratic function of the velocity, which depends linearly on the photon pressure and thus the laser power.
In relation to the laser power at the tracking detection point, this corresponds to a value of ∼ 12.5 mW. The measured hydrodynamic diameters dFaNTA-trans and dFaNTA-long (see Tab. S-1) correspond to the mean value of all measurements below the power threshold at the tracking detection point (below ∼12.5 mW in case of the 50 nm AuNp).
With the help of the theoretical known scattering and absorption cross section, it is possible to transfer max−out (photon pressure) to any other particle [see Tab. S-1 in SI 5], demonstrating that none of our measurements suffers from photon pressure.   With the help of the theoretical known scattering and absorption cross-section, it is possible to transfer max (heat) to any other particle [see Tab. S-1 in SI 5], demonstrating that none of our measurements suffers from thermal problems.

SI 8: Characterization of 7 nm gold nanoparticles
We would have liked to investigate whether we see anything beyond the demonstrated 9 nm AuNP. However, there are no stable and ultra-uniform AuNP available for 7 or 8 nm. The best specimen we could get are 7 nm AuNP with a CV of 10 % (AUCB7, nanoComposix). The specimen contains a significant amount of NP larger than 9 nm.
However, these particles are not distinguishable from the intended 7 nm NP by our method. Thus this specimen is not suitable to verify that we can see a 7 nm NP. In addition, we see agglomerations which did not occur with the other presented NP coming from the same manufacturer. Agglomerations can be found even after filtering the specimen by a 20 nm filter, indicating that the agglomerates reform quickly after the filter and cannot be avoided. In addition, the 20 nm filter blocks a significant amount of particles. Panalytical) were performed by successively diluting a highly concentrated solution of gold NP (physical diameter 50nm) and characterizing it in the Zetasizer (Fig. S-8 (a)).
In accordance with the discussion above, a very strong concentration dependence was found. For comparison, FaNTA experiments were performed with a specific solution (diluted with 0.1 % TWEEN20, concentration 10 −8 NP/ mL), which revealed an average diameter of 64.3 nm ( Fig. S-8 (b)). This clearly shows that diameter determination in DLS always requires a concentration series, which is not necessary when using FaNTA.  Fig. S-9. Experimentally, the light originating from the microchannel at the fiber output was imaged onto a Si-photodetector by means of a lens and a pinhole. A negligible variation in output power over this period is observed (standard deviation P = 0.0045), demonstrating the stability of intensity at the location of the NPs over a time period that is substantially longer than that of the experiments discussed here.
Note that the power variations are largely caused by the butt coupling between the delivery fiber and the ARE-fiber, which can be principally improved by an optimized design of the sample mount.  The data analysis yields a hydrodynamic diameter averaged over all ten   In the following, the latter model, which describes diffusion inside an infinitely extended cylinder, will be used to unravel the influence of the confinement effect on diffusion along the cylinder (i.e., fiber) axis (longitudinal direction). In all models, the crucial parameter is the ratio between the radii of the NP and the channel = / ( : radius of NP, : radius of channel). This dimensionless parameter allows calculating the resistance factor independent of the actual geometric dimensions of the system investigated.
The most important parameter in the work of J. M. Nitsche und G. Balgi 7 is the diffusion correction factor (Eq. 45 of the mentioned work) which is determined for various combinations of NP and channel radii (Fig. S-13(a)). This parameter is inverse proportional to the resistance factor ( = 1/ s ) and describes how the longitudinal free diffusion coefficient is changed by the confinement ( = � / ∞ ; measured and free diffusion coefficients: � and ∞ ). Clearly visible is a substantial deviation of the diffusion correction factor from unity, particularly in the case of small channel radii and large NPs, i.e., large values of . This effect can also be seen in  To demonstrate this influence, the diffusion of NPs in a circular geometry was simulated according to the parameters used in the experiments, and the corresponding hydrodynamic diameters were calculated. For each channel diameter, 50 simulations were performed, and the results were averaged to improve statistics ( Fig. S-14).

SI 19: Optimization of light incoupling
In the following, the light coupling procedure into the ARE is described: The optimization of the butt coupling is performed manually and includes the adjustment of the delivery fiber in -direction and the rotation of the SEF around its axis. Here, the delivery fiber is mounted on a fiber launching stage (MDE122, Elliot Scientific), ensuring a sufficient adjustment distance of 2 mm/axis with a resolution of 20 nm.
The SEF is located on a holder which allows manual rotation of the fiber, aiming to position the SEF in such a way that the contact point of ARE and jacket faces the objective. The light at the output of the ARE, i.e., SEF is used as an indicator for the quality of the alignment regarding exciting the fundamental ARE-mode and is imaged onto a camera chip using a lens. An example image of a measured profile of the fundamental mode is shown in Fig. S17. 20. SI 20: Description of z-score filtering The aim of the procedure is to sort out NPs that do not belong to the Ultra Uniform NP ensemble by means of a statistical evaluation. Here we use the so-called z-score, which describes the relationship of a value to the mean value of a group of values. If a NP has a z-score above a certain max , then it is most likely an outlier (agglomerate, dirt, etc.) and must be sorted out. Note that it is important to consider that the standard deviation in the context of NTA depends on trajectory length. The details of the data evaluation procedure are as follows: • The starting point is a matrix of the measured values with the following columns: NP-ID, diameter, diffusion coefficient, trajectory length, and brightness.
• This table is sorted with respect to trajectory length and thus accuracy. • As next step, the z-score of each trajectory is determined • Then the filtering according to the max is conducted.
• If at least one NP is filtered out, the entire analysis starts over until it converges and all NP fit the hypothesis of normal distribution.
In the current analysis, we use max = 2.576, which corresponds to one false event for 1000 events and 100 data points for each rolling window. If fewer trajectories are evaluated, the value is limited to the number of objects. A direct comparison of filtered and unfiltered data is shown in Fig. SI-18, revealing that statistical outliers can be effectively suppressed by this filtering procedure.